Dynamics and stability of weakly viscoelastic film flowing down a uniformly heated slippery incline

In this study, we investigate the stability of a thin viscoelastic fluid draining down a uniformly heated slippery inclined plane. A theoretical model is employed consisting of the Navier-Stokes equations coupled with the conservation equation for energy. We apply a Navier slip condition at the solid-liquid interface. To obtain the critical conditions for the onset of instability, we carry out a long-wave linear stability analysis within the Orr-Sommerfeld framework. Furthermore, we derive a first-order Benney-type evolution equation for the local film thickness to analyze the effect of long-wave instabilities. The results reveal that the slippery substrate destabilizes the liquid film flow. We find that the presence of the viscoelastic parameter and Marangoni number always promotes this destabilizing effect. We use the method of multiple scales to investigate the weakly nonlinear stability analysis of the flow which shows that there is a range of wave numbers with a supercritical bifurcation and a range of larger wave numbers with a subcritical bifurcation. The study interprets that the variation of Marangoni number, slip length and viscoelastic parameter have substantial effects on different stable or unstable zones. Different instability zones are also demarcated. Finally, the direct numerical simulations of the full thin-film model clearly demonstrate the role of the viscoelastic parameter, thermocapillary, and slip length. A good agreement between the linear stability analysis and the numerical simulations is found.